1. Regular Languages: Deterministic Finite Automata (DFA) 0 32 1 a a a a b b bb DFA = NFA (Non-deterministic) = REG {w  {a,b} * | # a W and # a W both even} aa abab abaaaabbb

2. Learning DFA’s: 2) MCA(S+) a S+ = (c, abc, ababc} 012345 678 abbc 9 bc a c Maximal Canonical Automaton

3. Learning DFA’s: 4) State Merging (Oncina,Vidal 92, Lang 98) a S+ = (c, abc, ababc} 012345 8 abbc 9 c c Evidence Driven State merging: 0,2

4. Learning DFA’s: 4) State Merging (Oncina,Vidal 92, Lang 98) a S+ = (c, abc, ababc} 01345 8 a bc 9 c c Evidence Driven State merging: 1,3 and 8,9 b

5. Learning DFA’s: 4) State Merging (Oncina,Vidal 92, Lang 98) a S+ = (c, abc, ababc} 0145 bc 9 c Evidence Driven State merging: 0,4 b

6. Learning DFA’s: 4) State Merging (Oncina,Vidal 92, Lang 98) a S+ = (c, abc, ababc} 015 c 9 c Evidence Driven State merging: 9,5 b

7. Learning DFA’s: 4) State Merging (Oncina,Vidal 92, Lang 98) a S+ = (c, abc, ababc} 01 9 c Evidence Driven State merging: 0,4 and 9,5 b

8. Learning DFA’s via evidence driven state merging  Input S+, S-  Output: DFA  1) Form MCA(S+) 2) Form PTA(S+) 3) Do until no merging is possible: - choose merging of two states - perform cascade of forced mergings to get a deterministic automaton - if resulting DFA accepts sentences of S- backtrack and choose another couple 4) End  Drawback: we need negative examples!!!

9. Learning DFA using only positive examples with MDL S+ = (c, cab, cabab, cababab, cababababab } c 012 b a c 0 L1 L2 b a Coding in bits: |L1|  5 log 2 (3+1) 2 log 2 (1+1) = 20 |L2|  5 log 2 (3+1) 2 log 2 (1+3) = 40 # arrows # letters Empty letter # states Outside world

10. Learning DFA using only positive examples with MDL S+ = (c, cab, cabab, cababab, cababababab } c 012 b a c 0 L1 L2 b a Coding in bits: |L1|  5 log 2 (3+1) 2 log 2 (1+1) = 20 |L2|  5 log 2 (3+1) 2 log 2 (1+3) = 40 But: L1 has 4 choices in state 0: |L1(S+)|= 26 log 2 4 = 52 L2 has 2 choices in state 1: |L2(S+)|= 16 log 2 2 = 16 |L2| + |L2(S+)|= 40 + 16 < |L1| + |L1(S+)|= 20 + 52 L2 is the better theory according to MDL

11. Learning DFA’s using only positive examples with MDL  Input S+  Output: DFA  1) Form MCA(S+) 2) Form DFA = PTA(S+) 3) Do until no merging is possible: - choose merging of two states - perform cascade of forced mergings to get a deterministic automaton DFA’ - if |DFA’|+|DFA’(S+)|  |DFA|+|DFA(S+)| backtrack and choose another couple 4) End  Drawback: Local minima!